Properties

Label 2352.4.a.cp.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(13.2349\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -18.9580 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -18.9580 q^{5} +9.00000 q^{9} -54.7308 q^{11} -62.0173 q^{13} -56.8741 q^{15} -122.439 q^{17} -12.5058 q^{19} +74.4391 q^{23} +234.406 q^{25} +27.0000 q^{27} -232.572 q^{29} +10.3677 q^{31} -164.192 q^{33} -245.987 q^{37} -186.052 q^{39} -238.653 q^{41} +92.9718 q^{43} -170.622 q^{45} -485.645 q^{47} -367.317 q^{51} -378.557 q^{53} +1037.59 q^{55} -37.5173 q^{57} -182.784 q^{59} -396.470 q^{61} +1175.73 q^{65} +261.239 q^{67} +223.317 q^{69} +874.523 q^{71} -152.406 q^{73} +703.219 q^{75} -573.357 q^{79} +81.0000 q^{81} +317.754 q^{83} +2321.20 q^{85} -697.717 q^{87} +95.0160 q^{89} +31.1032 q^{93} +237.084 q^{95} +1608.78 q^{97} -492.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} - 4q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} - 4q^{5} + 36q^{9} - 14q^{11} - 22q^{13} - 12q^{15} - 96q^{17} - 26q^{19} - 96q^{23} + 110q^{25} + 108q^{27} - 76q^{29} + 238q^{31} - 42q^{33} + 562q^{37} - 66q^{39} - 428q^{41} + 258q^{43} - 36q^{45} - 80q^{47} - 288q^{51} + 1476q^{55} - 78q^{57} + 262q^{59} + 276q^{61} + 2196q^{65} - 150q^{67} - 288q^{69} + 848q^{71} + 218q^{73} + 330q^{75} - 1762q^{79} + 324q^{81} + 3450q^{83} + 1452q^{85} - 228q^{87} + 344q^{89} + 714q^{93} - 2004q^{95} + 622q^{97} - 126q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −18.9580 −1.69566 −0.847828 0.530271i \(-0.822090\pi\)
−0.847828 + 0.530271i \(0.822090\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −54.7308 −1.50018 −0.750089 0.661337i \(-0.769989\pi\)
−0.750089 + 0.661337i \(0.769989\pi\)
\(12\) 0 0
\(13\) −62.0173 −1.32312 −0.661558 0.749894i \(-0.730104\pi\)
−0.661558 + 0.749894i \(0.730104\pi\)
\(14\) 0 0
\(15\) −56.8741 −0.978988
\(16\) 0 0
\(17\) −122.439 −1.74681 −0.873407 0.486991i \(-0.838095\pi\)
−0.873407 + 0.486991i \(0.838095\pi\)
\(18\) 0 0
\(19\) −12.5058 −0.151001 −0.0755004 0.997146i \(-0.524055\pi\)
−0.0755004 + 0.997146i \(0.524055\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 74.4391 0.674853 0.337427 0.941352i \(-0.390444\pi\)
0.337427 + 0.941352i \(0.390444\pi\)
\(24\) 0 0
\(25\) 234.406 1.87525
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −232.572 −1.48923 −0.744613 0.667496i \(-0.767366\pi\)
−0.744613 + 0.667496i \(0.767366\pi\)
\(30\) 0 0
\(31\) 10.3677 0.0600677 0.0300339 0.999549i \(-0.490438\pi\)
0.0300339 + 0.999549i \(0.490438\pi\)
\(32\) 0 0
\(33\) −164.192 −0.866128
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −245.987 −1.09297 −0.546486 0.837468i \(-0.684035\pi\)
−0.546486 + 0.837468i \(0.684035\pi\)
\(38\) 0 0
\(39\) −186.052 −0.763901
\(40\) 0 0
\(41\) −238.653 −0.909056 −0.454528 0.890733i \(-0.650192\pi\)
−0.454528 + 0.890733i \(0.650192\pi\)
\(42\) 0 0
\(43\) 92.9718 0.329722 0.164861 0.986317i \(-0.447282\pi\)
0.164861 + 0.986317i \(0.447282\pi\)
\(44\) 0 0
\(45\) −170.622 −0.565219
\(46\) 0 0
\(47\) −485.645 −1.50720 −0.753601 0.657332i \(-0.771685\pi\)
−0.753601 + 0.657332i \(0.771685\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −367.317 −1.00852
\(52\) 0 0
\(53\) −378.557 −0.981109 −0.490555 0.871410i \(-0.663206\pi\)
−0.490555 + 0.871410i \(0.663206\pi\)
\(54\) 0 0
\(55\) 1037.59 2.54379
\(56\) 0 0
\(57\) −37.5173 −0.0871804
\(58\) 0 0
\(59\) −182.784 −0.403329 −0.201664 0.979455i \(-0.564635\pi\)
−0.201664 + 0.979455i \(0.564635\pi\)
\(60\) 0 0
\(61\) −396.470 −0.832177 −0.416088 0.909324i \(-0.636599\pi\)
−0.416088 + 0.909324i \(0.636599\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1175.73 2.24355
\(66\) 0 0
\(67\) 261.239 0.476350 0.238175 0.971222i \(-0.423451\pi\)
0.238175 + 0.971222i \(0.423451\pi\)
\(68\) 0 0
\(69\) 223.317 0.389627
\(70\) 0 0
\(71\) 874.523 1.46179 0.730893 0.682492i \(-0.239104\pi\)
0.730893 + 0.682492i \(0.239104\pi\)
\(72\) 0 0
\(73\) −152.406 −0.244354 −0.122177 0.992508i \(-0.538988\pi\)
−0.122177 + 0.992508i \(0.538988\pi\)
\(74\) 0 0
\(75\) 703.219 1.08268
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −573.357 −0.816554 −0.408277 0.912858i \(-0.633870\pi\)
−0.408277 + 0.912858i \(0.633870\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 317.754 0.420217 0.210108 0.977678i \(-0.432618\pi\)
0.210108 + 0.977678i \(0.432618\pi\)
\(84\) 0 0
\(85\) 2321.20 2.96200
\(86\) 0 0
\(87\) −697.717 −0.859806
\(88\) 0 0
\(89\) 95.0160 0.113165 0.0565824 0.998398i \(-0.481980\pi\)
0.0565824 + 0.998398i \(0.481980\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 31.1032 0.0346801
\(94\) 0 0
\(95\) 237.084 0.256046
\(96\) 0 0
\(97\) 1608.78 1.68398 0.841992 0.539490i \(-0.181383\pi\)
0.841992 + 0.539490i \(0.181383\pi\)
\(98\) 0 0
\(99\) −492.577 −0.500059
\(100\) 0 0
\(101\) 783.043 0.771442 0.385721 0.922615i \(-0.373953\pi\)
0.385721 + 0.922615i \(0.373953\pi\)
\(102\) 0 0
\(103\) −1489.61 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −712.769 −0.643981 −0.321991 0.946743i \(-0.604352\pi\)
−0.321991 + 0.946743i \(0.604352\pi\)
\(108\) 0 0
\(109\) 1041.85 0.915513 0.457757 0.889078i \(-0.348653\pi\)
0.457757 + 0.889078i \(0.348653\pi\)
\(110\) 0 0
\(111\) −737.960 −0.631028
\(112\) 0 0
\(113\) 352.093 0.293116 0.146558 0.989202i \(-0.453181\pi\)
0.146558 + 0.989202i \(0.453181\pi\)
\(114\) 0 0
\(115\) −1411.22 −1.14432
\(116\) 0 0
\(117\) −558.156 −0.441039
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1664.46 1.25053
\(122\) 0 0
\(123\) −715.958 −0.524844
\(124\) 0 0
\(125\) −2074.13 −1.48413
\(126\) 0 0
\(127\) 1093.73 0.764194 0.382097 0.924122i \(-0.375202\pi\)
0.382097 + 0.924122i \(0.375202\pi\)
\(128\) 0 0
\(129\) 278.915 0.190365
\(130\) 0 0
\(131\) −2166.37 −1.44486 −0.722430 0.691444i \(-0.756975\pi\)
−0.722430 + 0.691444i \(0.756975\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −511.866 −0.326329
\(136\) 0 0
\(137\) −1964.92 −1.22536 −0.612681 0.790330i \(-0.709909\pi\)
−0.612681 + 0.790330i \(0.709909\pi\)
\(138\) 0 0
\(139\) −136.976 −0.0835840 −0.0417920 0.999126i \(-0.513307\pi\)
−0.0417920 + 0.999126i \(0.513307\pi\)
\(140\) 0 0
\(141\) −1456.93 −0.870184
\(142\) 0 0
\(143\) 3394.26 1.98491
\(144\) 0 0
\(145\) 4409.11 2.52522
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −280.105 −0.154007 −0.0770037 0.997031i \(-0.524535\pi\)
−0.0770037 + 0.997031i \(0.524535\pi\)
\(150\) 0 0
\(151\) −2194.13 −1.18249 −0.591245 0.806492i \(-0.701363\pi\)
−0.591245 + 0.806492i \(0.701363\pi\)
\(152\) 0 0
\(153\) −1101.95 −0.582271
\(154\) 0 0
\(155\) −196.552 −0.101854
\(156\) 0 0
\(157\) 220.818 0.112250 0.0561248 0.998424i \(-0.482126\pi\)
0.0561248 + 0.998424i \(0.482126\pi\)
\(158\) 0 0
\(159\) −1135.67 −0.566444
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1476.86 −0.709674 −0.354837 0.934928i \(-0.615464\pi\)
−0.354837 + 0.934928i \(0.615464\pi\)
\(164\) 0 0
\(165\) 3112.76 1.46866
\(166\) 0 0
\(167\) 2197.66 1.01832 0.509162 0.860671i \(-0.329955\pi\)
0.509162 + 0.860671i \(0.329955\pi\)
\(168\) 0 0
\(169\) 1649.15 0.750635
\(170\) 0 0
\(171\) −112.552 −0.0503336
\(172\) 0 0
\(173\) −2091.61 −0.919201 −0.459601 0.888126i \(-0.652007\pi\)
−0.459601 + 0.888126i \(0.652007\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −548.351 −0.232862
\(178\) 0 0
\(179\) −2334.54 −0.974813 −0.487406 0.873175i \(-0.662057\pi\)
−0.487406 + 0.873175i \(0.662057\pi\)
\(180\) 0 0
\(181\) −1758.40 −0.722105 −0.361053 0.932545i \(-0.617583\pi\)
−0.361053 + 0.932545i \(0.617583\pi\)
\(182\) 0 0
\(183\) −1189.41 −0.480457
\(184\) 0 0
\(185\) 4663.42 1.85330
\(186\) 0 0
\(187\) 6701.19 2.62053
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3700.68 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(192\) 0 0
\(193\) 2708.45 1.01015 0.505073 0.863077i \(-0.331466\pi\)
0.505073 + 0.863077i \(0.331466\pi\)
\(194\) 0 0
\(195\) 3527.18 1.29531
\(196\) 0 0
\(197\) −160.686 −0.0581138 −0.0290569 0.999578i \(-0.509250\pi\)
−0.0290569 + 0.999578i \(0.509250\pi\)
\(198\) 0 0
\(199\) −2065.99 −0.735951 −0.367976 0.929835i \(-0.619949\pi\)
−0.367976 + 0.929835i \(0.619949\pi\)
\(200\) 0 0
\(201\) 783.717 0.275021
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4524.38 1.54145
\(206\) 0 0
\(207\) 669.952 0.224951
\(208\) 0 0
\(209\) 684.450 0.226528
\(210\) 0 0
\(211\) 1007.12 0.328592 0.164296 0.986411i \(-0.447465\pi\)
0.164296 + 0.986411i \(0.447465\pi\)
\(212\) 0 0
\(213\) 2623.57 0.843962
\(214\) 0 0
\(215\) −1762.56 −0.559096
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −457.219 −0.141078
\(220\) 0 0
\(221\) 7593.34 2.31124
\(222\) 0 0
\(223\) 1644.83 0.493929 0.246964 0.969025i \(-0.420567\pi\)
0.246964 + 0.969025i \(0.420567\pi\)
\(224\) 0 0
\(225\) 2109.66 0.625084
\(226\) 0 0
\(227\) 318.874 0.0932354 0.0466177 0.998913i \(-0.485156\pi\)
0.0466177 + 0.998913i \(0.485156\pi\)
\(228\) 0 0
\(229\) −2536.11 −0.731837 −0.365918 0.930647i \(-0.619245\pi\)
−0.365918 + 0.930647i \(0.619245\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2332.81 0.655913 0.327957 0.944693i \(-0.393640\pi\)
0.327957 + 0.944693i \(0.393640\pi\)
\(234\) 0 0
\(235\) 9206.86 2.55570
\(236\) 0 0
\(237\) −1720.07 −0.471438
\(238\) 0 0
\(239\) −2713.85 −0.734495 −0.367248 0.930123i \(-0.619700\pi\)
−0.367248 + 0.930123i \(0.619700\pi\)
\(240\) 0 0
\(241\) 4174.59 1.11580 0.557902 0.829907i \(-0.311606\pi\)
0.557902 + 0.829907i \(0.311606\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 775.573 0.199792
\(248\) 0 0
\(249\) 953.261 0.242612
\(250\) 0 0
\(251\) 6123.58 1.53991 0.769954 0.638099i \(-0.220279\pi\)
0.769954 + 0.638099i \(0.220279\pi\)
\(252\) 0 0
\(253\) −4074.11 −1.01240
\(254\) 0 0
\(255\) 6963.61 1.71011
\(256\) 0 0
\(257\) −4633.91 −1.12473 −0.562365 0.826889i \(-0.690108\pi\)
−0.562365 + 0.826889i \(0.690108\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2093.15 −0.496409
\(262\) 0 0
\(263\) −3283.23 −0.769781 −0.384891 0.922962i \(-0.625761\pi\)
−0.384891 + 0.922962i \(0.625761\pi\)
\(264\) 0 0
\(265\) 7176.69 1.66362
\(266\) 0 0
\(267\) 285.048 0.0653358
\(268\) 0 0
\(269\) 798.985 0.181097 0.0905483 0.995892i \(-0.471138\pi\)
0.0905483 + 0.995892i \(0.471138\pi\)
\(270\) 0 0
\(271\) −6706.07 −1.50319 −0.751596 0.659624i \(-0.770716\pi\)
−0.751596 + 0.659624i \(0.770716\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12829.3 −2.81321
\(276\) 0 0
\(277\) 1162.27 0.252108 0.126054 0.992023i \(-0.459769\pi\)
0.126054 + 0.992023i \(0.459769\pi\)
\(278\) 0 0
\(279\) 93.3096 0.0200226
\(280\) 0 0
\(281\) 2718.17 0.577055 0.288527 0.957472i \(-0.406834\pi\)
0.288527 + 0.957472i \(0.406834\pi\)
\(282\) 0 0
\(283\) −3009.48 −0.632138 −0.316069 0.948736i \(-0.602363\pi\)
−0.316069 + 0.948736i \(0.602363\pi\)
\(284\) 0 0
\(285\) 711.253 0.147828
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10078.3 2.05136
\(290\) 0 0
\(291\) 4826.33 0.972249
\(292\) 0 0
\(293\) −4209.15 −0.839252 −0.419626 0.907697i \(-0.637839\pi\)
−0.419626 + 0.907697i \(0.637839\pi\)
\(294\) 0 0
\(295\) 3465.21 0.683907
\(296\) 0 0
\(297\) −1477.73 −0.288709
\(298\) 0 0
\(299\) −4616.51 −0.892909
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2349.13 0.445392
\(304\) 0 0
\(305\) 7516.28 1.41109
\(306\) 0 0
\(307\) −3114.82 −0.579063 −0.289531 0.957169i \(-0.593499\pi\)
−0.289531 + 0.957169i \(0.593499\pi\)
\(308\) 0 0
\(309\) −4468.83 −0.822727
\(310\) 0 0
\(311\) −8487.24 −1.54748 −0.773741 0.633501i \(-0.781617\pi\)
−0.773741 + 0.633501i \(0.781617\pi\)
\(312\) 0 0
\(313\) 4954.91 0.894786 0.447393 0.894337i \(-0.352352\pi\)
0.447393 + 0.894337i \(0.352352\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5608.79 −0.993757 −0.496879 0.867820i \(-0.665521\pi\)
−0.496879 + 0.867820i \(0.665521\pi\)
\(318\) 0 0
\(319\) 12728.9 2.23411
\(320\) 0 0
\(321\) −2138.31 −0.371803
\(322\) 0 0
\(323\) 1531.19 0.263770
\(324\) 0 0
\(325\) −14537.3 −2.48117
\(326\) 0 0
\(327\) 3125.54 0.528572
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6609.49 1.09755 0.548777 0.835969i \(-0.315094\pi\)
0.548777 + 0.835969i \(0.315094\pi\)
\(332\) 0 0
\(333\) −2213.88 −0.364324
\(334\) 0 0
\(335\) −4952.57 −0.807725
\(336\) 0 0
\(337\) 11455.5 1.85170 0.925850 0.377891i \(-0.123350\pi\)
0.925850 + 0.377891i \(0.123350\pi\)
\(338\) 0 0
\(339\) 1056.28 0.169231
\(340\) 0 0
\(341\) −567.434 −0.0901123
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4233.65 −0.660673
\(346\) 0 0
\(347\) 6044.15 0.935064 0.467532 0.883976i \(-0.345143\pi\)
0.467532 + 0.883976i \(0.345143\pi\)
\(348\) 0 0
\(349\) 5487.93 0.841726 0.420863 0.907124i \(-0.361727\pi\)
0.420863 + 0.907124i \(0.361727\pi\)
\(350\) 0 0
\(351\) −1674.47 −0.254634
\(352\) 0 0
\(353\) −6880.15 −1.03738 −0.518688 0.854964i \(-0.673579\pi\)
−0.518688 + 0.854964i \(0.673579\pi\)
\(354\) 0 0
\(355\) −16579.2 −2.47869
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3938.34 0.578990 0.289495 0.957180i \(-0.406513\pi\)
0.289495 + 0.957180i \(0.406513\pi\)
\(360\) 0 0
\(361\) −6702.61 −0.977199
\(362\) 0 0
\(363\) 4993.38 0.721996
\(364\) 0 0
\(365\) 2889.32 0.414340
\(366\) 0 0
\(367\) 1439.63 0.204763 0.102381 0.994745i \(-0.467354\pi\)
0.102381 + 0.994745i \(0.467354\pi\)
\(368\) 0 0
\(369\) −2147.87 −0.303019
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2494.11 −0.346220 −0.173110 0.984903i \(-0.555382\pi\)
−0.173110 + 0.984903i \(0.555382\pi\)
\(374\) 0 0
\(375\) −6222.39 −0.856861
\(376\) 0 0
\(377\) 14423.5 1.97042
\(378\) 0 0
\(379\) 1309.25 0.177445 0.0887225 0.996056i \(-0.471722\pi\)
0.0887225 + 0.996056i \(0.471722\pi\)
\(380\) 0 0
\(381\) 3281.18 0.441208
\(382\) 0 0
\(383\) 6961.80 0.928803 0.464402 0.885625i \(-0.346269\pi\)
0.464402 + 0.885625i \(0.346269\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 836.746 0.109907
\(388\) 0 0
\(389\) 6353.45 0.828104 0.414052 0.910253i \(-0.364113\pi\)
0.414052 + 0.910253i \(0.364113\pi\)
\(390\) 0 0
\(391\) −9114.25 −1.17884
\(392\) 0 0
\(393\) −6499.11 −0.834191
\(394\) 0 0
\(395\) 10869.7 1.38459
\(396\) 0 0
\(397\) −11505.4 −1.45451 −0.727254 0.686368i \(-0.759204\pi\)
−0.727254 + 0.686368i \(0.759204\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1653.33 0.205893 0.102947 0.994687i \(-0.467173\pi\)
0.102947 + 0.994687i \(0.467173\pi\)
\(402\) 0 0
\(403\) −642.979 −0.0794766
\(404\) 0 0
\(405\) −1535.60 −0.188406
\(406\) 0 0
\(407\) 13463.0 1.63965
\(408\) 0 0
\(409\) 4894.83 0.591769 0.295885 0.955224i \(-0.404386\pi\)
0.295885 + 0.955224i \(0.404386\pi\)
\(410\) 0 0
\(411\) −5894.77 −0.707463
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6023.98 −0.712544
\(416\) 0 0
\(417\) −410.929 −0.0482573
\(418\) 0 0
\(419\) −265.504 −0.0309563 −0.0154782 0.999880i \(-0.504927\pi\)
−0.0154782 + 0.999880i \(0.504927\pi\)
\(420\) 0 0
\(421\) 11136.8 1.28925 0.644623 0.764500i \(-0.277014\pi\)
0.644623 + 0.764500i \(0.277014\pi\)
\(422\) 0 0
\(423\) −4370.80 −0.502401
\(424\) 0 0
\(425\) −28700.5 −3.27572
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 10182.8 1.14599
\(430\) 0 0
\(431\) −5193.41 −0.580413 −0.290206 0.956964i \(-0.593724\pi\)
−0.290206 + 0.956964i \(0.593724\pi\)
\(432\) 0 0
\(433\) −6314.17 −0.700785 −0.350392 0.936603i \(-0.613952\pi\)
−0.350392 + 0.936603i \(0.613952\pi\)
\(434\) 0 0
\(435\) 13227.3 1.45794
\(436\) 0 0
\(437\) −930.917 −0.101903
\(438\) 0 0
\(439\) −15423.3 −1.67680 −0.838399 0.545056i \(-0.816508\pi\)
−0.838399 + 0.545056i \(0.816508\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8706.11 0.933725 0.466862 0.884330i \(-0.345384\pi\)
0.466862 + 0.884330i \(0.345384\pi\)
\(444\) 0 0
\(445\) −1801.31 −0.191889
\(446\) 0 0
\(447\) −840.315 −0.0889162
\(448\) 0 0
\(449\) 5495.91 0.577657 0.288829 0.957381i \(-0.406734\pi\)
0.288829 + 0.957381i \(0.406734\pi\)
\(450\) 0 0
\(451\) 13061.7 1.36375
\(452\) 0 0
\(453\) −6582.40 −0.682711
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11357.3 −1.16252 −0.581258 0.813719i \(-0.697439\pi\)
−0.581258 + 0.813719i \(0.697439\pi\)
\(458\) 0 0
\(459\) −3305.86 −0.336175
\(460\) 0 0
\(461\) −14514.2 −1.46637 −0.733184 0.680030i \(-0.761967\pi\)
−0.733184 + 0.680030i \(0.761967\pi\)
\(462\) 0 0
\(463\) −9971.00 −1.00085 −0.500423 0.865781i \(-0.666822\pi\)
−0.500423 + 0.865781i \(0.666822\pi\)
\(464\) 0 0
\(465\) −589.655 −0.0588056
\(466\) 0 0
\(467\) −397.477 −0.0393856 −0.0196928 0.999806i \(-0.506269\pi\)
−0.0196928 + 0.999806i \(0.506269\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 662.454 0.0648074
\(472\) 0 0
\(473\) −5088.42 −0.494642
\(474\) 0 0
\(475\) −2931.43 −0.283165
\(476\) 0 0
\(477\) −3407.01 −0.327036
\(478\) 0 0
\(479\) −14060.4 −1.34120 −0.670602 0.741818i \(-0.733964\pi\)
−0.670602 + 0.741818i \(0.733964\pi\)
\(480\) 0 0
\(481\) 15255.4 1.44613
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −30499.2 −2.85546
\(486\) 0 0
\(487\) 13534.7 1.25937 0.629687 0.776849i \(-0.283183\pi\)
0.629687 + 0.776849i \(0.283183\pi\)
\(488\) 0 0
\(489\) −4430.59 −0.409730
\(490\) 0 0
\(491\) 8693.29 0.799028 0.399514 0.916727i \(-0.369179\pi\)
0.399514 + 0.916727i \(0.369179\pi\)
\(492\) 0 0
\(493\) 28475.9 2.60140
\(494\) 0 0
\(495\) 9338.29 0.847929
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4017.93 −0.360455 −0.180228 0.983625i \(-0.557683\pi\)
−0.180228 + 0.983625i \(0.557683\pi\)
\(500\) 0 0
\(501\) 6592.98 0.587929
\(502\) 0 0
\(503\) 52.2455 0.00463124 0.00231562 0.999997i \(-0.499263\pi\)
0.00231562 + 0.999997i \(0.499263\pi\)
\(504\) 0 0
\(505\) −14844.9 −1.30810
\(506\) 0 0
\(507\) 4947.44 0.433379
\(508\) 0 0
\(509\) −9239.10 −0.804550 −0.402275 0.915519i \(-0.631780\pi\)
−0.402275 + 0.915519i \(0.631780\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −337.655 −0.0290601
\(514\) 0 0
\(515\) 28240.0 2.41632
\(516\) 0 0
\(517\) 26579.7 2.26107
\(518\) 0 0
\(519\) −6274.82 −0.530701
\(520\) 0 0
\(521\) −14973.4 −1.25911 −0.629555 0.776956i \(-0.716763\pi\)
−0.629555 + 0.776956i \(0.716763\pi\)
\(522\) 0 0
\(523\) 13603.5 1.13736 0.568681 0.822558i \(-0.307454\pi\)
0.568681 + 0.822558i \(0.307454\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1269.42 −0.104927
\(528\) 0 0
\(529\) −6625.82 −0.544573
\(530\) 0 0
\(531\) −1645.05 −0.134443
\(532\) 0 0
\(533\) 14800.6 1.20279
\(534\) 0 0
\(535\) 13512.7 1.09197
\(536\) 0 0
\(537\) −7003.61 −0.562808
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −16886.4 −1.34196 −0.670981 0.741474i \(-0.734127\pi\)
−0.670981 + 0.741474i \(0.734127\pi\)
\(542\) 0 0
\(543\) −5275.21 −0.416908
\(544\) 0 0
\(545\) −19751.4 −1.55240
\(546\) 0 0
\(547\) 5987.25 0.468001 0.234000 0.972237i \(-0.424818\pi\)
0.234000 + 0.972237i \(0.424818\pi\)
\(548\) 0 0
\(549\) −3568.23 −0.277392
\(550\) 0 0
\(551\) 2908.49 0.224875
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 13990.3 1.07001
\(556\) 0 0
\(557\) 2238.71 0.170300 0.0851499 0.996368i \(-0.472863\pi\)
0.0851499 + 0.996368i \(0.472863\pi\)
\(558\) 0 0
\(559\) −5765.86 −0.436261
\(560\) 0 0
\(561\) 20103.6 1.51297
\(562\) 0 0
\(563\) 8452.84 0.632762 0.316381 0.948632i \(-0.397532\pi\)
0.316381 + 0.948632i \(0.397532\pi\)
\(564\) 0 0
\(565\) −6674.98 −0.497024
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14953.6 1.10173 0.550866 0.834594i \(-0.314298\pi\)
0.550866 + 0.834594i \(0.314298\pi\)
\(570\) 0 0
\(571\) −16021.8 −1.17424 −0.587122 0.809498i \(-0.699739\pi\)
−0.587122 + 0.809498i \(0.699739\pi\)
\(572\) 0 0
\(573\) 11102.0 0.809414
\(574\) 0 0
\(575\) 17449.0 1.26552
\(576\) 0 0
\(577\) 16113.1 1.16256 0.581279 0.813704i \(-0.302552\pi\)
0.581279 + 0.813704i \(0.302552\pi\)
\(578\) 0 0
\(579\) 8125.34 0.583208
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20718.7 1.47184
\(584\) 0 0
\(585\) 10581.5 0.747850
\(586\) 0 0
\(587\) −9552.04 −0.671644 −0.335822 0.941925i \(-0.609014\pi\)
−0.335822 + 0.941925i \(0.609014\pi\)
\(588\) 0 0
\(589\) −129.656 −0.00907028
\(590\) 0 0
\(591\) −482.058 −0.0335520
\(592\) 0 0
\(593\) 24167.8 1.67361 0.836807 0.547499i \(-0.184420\pi\)
0.836807 + 0.547499i \(0.184420\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6197.98 −0.424902
\(598\) 0 0
\(599\) −821.331 −0.0560245 −0.0280122 0.999608i \(-0.508918\pi\)
−0.0280122 + 0.999608i \(0.508918\pi\)
\(600\) 0 0
\(601\) 14674.7 0.995996 0.497998 0.867178i \(-0.334069\pi\)
0.497998 + 0.867178i \(0.334069\pi\)
\(602\) 0 0
\(603\) 2351.15 0.158783
\(604\) 0 0
\(605\) −31554.9 −2.12048
\(606\) 0 0
\(607\) 5083.12 0.339897 0.169949 0.985453i \(-0.445640\pi\)
0.169949 + 0.985453i \(0.445640\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 30118.4 1.99420
\(612\) 0 0
\(613\) −22202.0 −1.46286 −0.731428 0.681919i \(-0.761146\pi\)
−0.731428 + 0.681919i \(0.761146\pi\)
\(614\) 0 0
\(615\) 13573.1 0.889954
\(616\) 0 0
\(617\) −14990.6 −0.978121 −0.489060 0.872250i \(-0.662660\pi\)
−0.489060 + 0.872250i \(0.662660\pi\)
\(618\) 0 0
\(619\) 3073.02 0.199540 0.0997700 0.995011i \(-0.468189\pi\)
0.0997700 + 0.995011i \(0.468189\pi\)
\(620\) 0 0
\(621\) 2009.86 0.129876
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 10020.6 0.641317
\(626\) 0 0
\(627\) 2053.35 0.130786
\(628\) 0 0
\(629\) 30118.4 1.90922
\(630\) 0 0
\(631\) −26012.7 −1.64113 −0.820563 0.571556i \(-0.806340\pi\)
−0.820563 + 0.571556i \(0.806340\pi\)
\(632\) 0 0
\(633\) 3021.36 0.189713
\(634\) 0 0
\(635\) −20734.9 −1.29581
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7870.70 0.487262
\(640\) 0 0
\(641\) 8032.23 0.494936 0.247468 0.968896i \(-0.420401\pi\)
0.247468 + 0.968896i \(0.420401\pi\)
\(642\) 0 0
\(643\) −24887.7 −1.52640 −0.763200 0.646162i \(-0.776373\pi\)
−0.763200 + 0.646162i \(0.776373\pi\)
\(644\) 0 0
\(645\) −5287.68 −0.322794
\(646\) 0 0
\(647\) −21084.3 −1.28116 −0.640580 0.767891i \(-0.721306\pi\)
−0.640580 + 0.767891i \(0.721306\pi\)
\(648\) 0 0
\(649\) 10003.9 0.605065
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13166.3 0.789033 0.394516 0.918889i \(-0.370912\pi\)
0.394516 + 0.918889i \(0.370912\pi\)
\(654\) 0 0
\(655\) 41070.1 2.44999
\(656\) 0 0
\(657\) −1371.66 −0.0814513
\(658\) 0 0
\(659\) 13903.4 0.821851 0.410926 0.911669i \(-0.365206\pi\)
0.410926 + 0.911669i \(0.365206\pi\)
\(660\) 0 0
\(661\) −6306.61 −0.371102 −0.185551 0.982635i \(-0.559407\pi\)
−0.185551 + 0.982635i \(0.559407\pi\)
\(662\) 0 0
\(663\) 22780.0 1.33439
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −17312.5 −1.00501
\(668\) 0 0
\(669\) 4934.50 0.285170
\(670\) 0 0
\(671\) 21699.1 1.24841
\(672\) 0 0
\(673\) −24407.6 −1.39798 −0.698992 0.715129i \(-0.746368\pi\)
−0.698992 + 0.715129i \(0.746368\pi\)
\(674\) 0 0
\(675\) 6328.97 0.360892
\(676\) 0 0
\(677\) 31080.3 1.76442 0.882209 0.470858i \(-0.156055\pi\)
0.882209 + 0.470858i \(0.156055\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 956.623 0.0538295
\(682\) 0 0
\(683\) −28759.5 −1.61120 −0.805601 0.592458i \(-0.798157\pi\)
−0.805601 + 0.592458i \(0.798157\pi\)
\(684\) 0 0
\(685\) 37251.0 2.07779
\(686\) 0 0
\(687\) −7608.32 −0.422526
\(688\) 0 0
\(689\) 23477.1 1.29812
\(690\) 0 0
\(691\) 24895.4 1.37057 0.685287 0.728273i \(-0.259677\pi\)
0.685287 + 0.728273i \(0.259677\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2596.80 0.141730
\(696\) 0 0
\(697\) 29220.4 1.58795
\(698\) 0 0
\(699\) 6998.44 0.378692
\(700\) 0 0
\(701\) −1702.74 −0.0917427 −0.0458714 0.998947i \(-0.514606\pi\)
−0.0458714 + 0.998947i \(0.514606\pi\)
\(702\) 0 0
\(703\) 3076.25 0.165040
\(704\) 0 0
\(705\) 27620.6 1.47553
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6523.21 0.345535 0.172767 0.984963i \(-0.444729\pi\)
0.172767 + 0.984963i \(0.444729\pi\)
\(710\) 0 0
\(711\) −5160.22 −0.272185
\(712\) 0 0
\(713\) 771.765 0.0405369
\(714\) 0 0
\(715\) −64348.4 −3.36572
\(716\) 0 0
\(717\) −8141.55 −0.424061
\(718\) 0 0
\(719\) −25254.4 −1.30992 −0.654959 0.755664i \(-0.727314\pi\)
−0.654959 + 0.755664i \(0.727314\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 12523.8 0.644210
\(724\) 0 0
\(725\) −54516.4 −2.79268
\(726\) 0 0
\(727\) −27964.9 −1.42663 −0.713316 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −11383.4 −0.575964
\(732\) 0 0
\(733\) 9294.38 0.468344 0.234172 0.972195i \(-0.424762\pi\)
0.234172 + 0.972195i \(0.424762\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14297.8 −0.714609
\(738\) 0 0
\(739\) −18291.2 −0.910490 −0.455245 0.890366i \(-0.650448\pi\)
−0.455245 + 0.890366i \(0.650448\pi\)
\(740\) 0 0
\(741\) 2326.72 0.115350
\(742\) 0 0
\(743\) −14742.4 −0.727921 −0.363960 0.931414i \(-0.618576\pi\)
−0.363960 + 0.931414i \(0.618576\pi\)
\(744\) 0 0
\(745\) 5310.24 0.261144
\(746\) 0 0
\(747\) 2859.78 0.140072
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28463.4 −1.38301 −0.691507 0.722370i \(-0.743053\pi\)
−0.691507 + 0.722370i \(0.743053\pi\)
\(752\) 0 0
\(753\) 18370.7 0.889067
\(754\) 0 0
\(755\) 41596.4 2.00510
\(756\) 0 0
\(757\) −20336.7 −0.976422 −0.488211 0.872726i \(-0.662350\pi\)
−0.488211 + 0.872726i \(0.662350\pi\)
\(758\) 0 0
\(759\) −12222.3 −0.584509
\(760\) 0 0
\(761\) −39581.6 −1.88546 −0.942728 0.333564i \(-0.891749\pi\)
−0.942728 + 0.333564i \(0.891749\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 20890.8 0.987332
\(766\) 0 0
\(767\) 11335.7 0.533650
\(768\) 0 0
\(769\) −10580.3 −0.496146 −0.248073 0.968741i \(-0.579797\pi\)
−0.248073 + 0.968741i \(0.579797\pi\)
\(770\) 0 0
\(771\) −13901.7 −0.649363
\(772\) 0 0
\(773\) −25922.2 −1.20615 −0.603077 0.797683i \(-0.706059\pi\)
−0.603077 + 0.797683i \(0.706059\pi\)
\(774\) 0 0
\(775\) 2430.26 0.112642
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2984.53 0.137268
\(780\) 0 0
\(781\) −47863.3 −2.19294
\(782\) 0 0
\(783\) −6279.45 −0.286602
\(784\) 0 0
\(785\) −4186.27 −0.190337
\(786\) 0 0
\(787\) 17220.0 0.779958 0.389979 0.920824i \(-0.372482\pi\)
0.389979 + 0.920824i \(0.372482\pi\)
\(788\) 0 0
\(789\) −9849.68 −0.444433
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24588.0 1.10107
\(794\) 0 0
\(795\) 21530.1 0.960494
\(796\) 0 0
\(797\) 6275.52 0.278909 0.139454 0.990228i \(-0.455465\pi\)
0.139454 + 0.990228i \(0.455465\pi\)
\(798\) 0 0
\(799\) 59461.9 2.63280
\(800\) 0 0
\(801\) 855.144 0.0377216
\(802\) 0 0
\(803\) 8341.33 0.366574
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2396.96 0.104556
\(808\) 0 0
\(809\) −4604.03 −0.200085 −0.100043 0.994983i \(-0.531898\pi\)
−0.100043 + 0.994983i \(0.531898\pi\)
\(810\) 0 0
\(811\) 5104.36 0.221009 0.110505 0.993876i \(-0.464753\pi\)
0.110505 + 0.993876i \(0.464753\pi\)
\(812\) 0 0
\(813\) −20118.2 −0.867868
\(814\) 0 0
\(815\) 27998.4 1.20336
\(816\) 0 0
\(817\) −1162.68 −0.0497884
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45635.4 −1.93993 −0.969967 0.243237i \(-0.921791\pi\)
−0.969967 + 0.243237i \(0.921791\pi\)
\(822\) 0 0
\(823\) 39377.1 1.66780 0.833901 0.551915i \(-0.186103\pi\)
0.833901 + 0.551915i \(0.186103\pi\)
\(824\) 0 0
\(825\) −38487.8 −1.62421
\(826\) 0 0
\(827\) 30916.3 1.29996 0.649978 0.759953i \(-0.274778\pi\)
0.649978 + 0.759953i \(0.274778\pi\)
\(828\) 0 0
\(829\) 2543.33 0.106554 0.0532771 0.998580i \(-0.483033\pi\)
0.0532771 + 0.998580i \(0.483033\pi\)
\(830\) 0 0
\(831\) 3486.81 0.145555
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −41663.3 −1.72673
\(836\) 0 0
\(837\) 279.929 0.0115600
\(838\) 0 0
\(839\) −7930.82 −0.326343 −0.163172 0.986598i \(-0.552172\pi\)
−0.163172 + 0.986598i \(0.552172\pi\)
\(840\) 0 0
\(841\) 29700.8 1.21780
\(842\) 0 0
\(843\) 8154.51 0.333163
\(844\) 0 0
\(845\) −31264.5 −1.27282
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9028.45 −0.364965
\(850\) 0 0
\(851\) −18311.0 −0.737595
\(852\) 0 0
\(853\) −41983.2 −1.68520 −0.842601 0.538538i \(-0.818977\pi\)
−0.842601 + 0.538538i \(0.818977\pi\)
\(854\) 0 0
\(855\) 2133.76 0.0853485
\(856\) 0 0
\(857\) 11854.3 0.472504 0.236252 0.971692i \(-0.424081\pi\)
0.236252 + 0.971692i \(0.424081\pi\)
\(858\) 0 0
\(859\) −8113.88 −0.322284 −0.161142 0.986931i \(-0.551518\pi\)
−0.161142 + 0.986931i \(0.551518\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45817.0 −1.80722 −0.903609 0.428358i \(-0.859092\pi\)
−0.903609 + 0.428358i \(0.859092\pi\)
\(864\) 0 0
\(865\) 39652.7 1.55865
\(866\) 0 0
\(867\) 30235.0 1.18435
\(868\) 0 0
\(869\) 31380.3 1.22498
\(870\) 0 0
\(871\) −16201.3 −0.630266
\(872\) 0 0
\(873\) 14479.0 0.561328
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31813.0 1.22491 0.612456 0.790504i \(-0.290182\pi\)
0.612456 + 0.790504i \(0.290182\pi\)
\(878\) 0 0
\(879\) −12627.4 −0.484543
\(880\) 0 0
\(881\) −43551.6 −1.66548 −0.832742 0.553661i \(-0.813230\pi\)
−0.832742 + 0.553661i \(0.813230\pi\)
\(882\) 0 0
\(883\) −40645.1 −1.54906 −0.774528 0.632540i \(-0.782012\pi\)
−0.774528 + 0.632540i \(0.782012\pi\)
\(884\) 0 0
\(885\) 10395.6 0.394854
\(886\) 0 0
\(887\) 34028.2 1.28811 0.644056 0.764978i \(-0.277250\pi\)
0.644056 + 0.764978i \(0.277250\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4433.20 −0.166686
\(892\) 0 0
\(893\) 6073.35 0.227589
\(894\) 0 0
\(895\) 44258.2 1.65295
\(896\) 0 0
\(897\) −13849.5 −0.515521
\(898\) 0 0
\(899\) −2411.25 −0.0894545
\(900\) 0 0
\(901\) 46350.2 1.71382
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33335.8 1.22444
\(906\) 0 0
\(907\) −50804.7 −1.85992 −0.929958 0.367666i \(-0.880157\pi\)
−0.929958 + 0.367666i \(0.880157\pi\)
\(908\) 0 0
\(909\) 7047.38 0.257147
\(910\) 0 0
\(911\) −28738.3 −1.04516 −0.522582 0.852589i \(-0.675031\pi\)
−0.522582 + 0.852589i \(0.675031\pi\)
\(912\) 0 0
\(913\) −17390.9 −0.630400
\(914\) 0 0
\(915\) 22548.9 0.814691
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 54742.2 1.96494 0.982470 0.186420i \(-0.0596886\pi\)
0.982470 + 0.186420i \(0.0596886\pi\)
\(920\) 0 0
\(921\) −9344.46 −0.334322
\(922\) 0 0
\(923\) −54235.5 −1.93411
\(924\) 0 0
\(925\) −57660.9 −2.04960
\(926\) 0 0
\(927\) −13406.5 −0.475002
\(928\) 0 0
\(929\) −43775.8 −1.54600 −0.773002 0.634403i \(-0.781246\pi\)
−0.773002 + 0.634403i \(0.781246\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −25461.7 −0.893440
\(934\) 0 0
\(935\) −127041. −4.44352
\(936\) 0 0
\(937\) −35090.9 −1.22345 −0.611724 0.791071i \(-0.709524\pi\)
−0.611724 + 0.791071i \(0.709524\pi\)
\(938\) 0 0
\(939\) 14864.7 0.516605
\(940\) 0 0
\(941\) 31784.8 1.10112 0.550560 0.834795i \(-0.314414\pi\)
0.550560 + 0.834795i \(0.314414\pi\)
\(942\) 0 0
\(943\) −17765.1 −0.613479
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −56500.9 −1.93879 −0.969394 0.245510i \(-0.921045\pi\)
−0.969394 + 0.245510i \(0.921045\pi\)
\(948\) 0 0
\(949\) 9451.84 0.323308
\(950\) 0 0
\(951\) −16826.4 −0.573746
\(952\) 0 0
\(953\) 36669.1 1.24641 0.623204 0.782059i \(-0.285831\pi\)
0.623204 + 0.782059i \(0.285831\pi\)
\(954\) 0 0
\(955\) −70157.5 −2.37722
\(956\) 0 0
\(957\) 38186.6 1.28986
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29683.5 −0.996392
\(962\) 0 0
\(963\) −6414.92 −0.214660
\(964\) 0 0
\(965\) −51346.8 −1.71286
\(966\) 0 0
\(967\) 12012.4 0.399474 0.199737 0.979850i \(-0.435991\pi\)
0.199737 + 0.979850i \(0.435991\pi\)
\(968\) 0 0
\(969\) 4593.58 0.152288
\(970\) 0 0
\(971\) −37296.2 −1.23264 −0.616320 0.787496i \(-0.711377\pi\)
−0.616320 + 0.787496i \(0.711377\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −43611.8 −1.43251
\(976\) 0 0
\(977\) 9788.45 0.320532 0.160266 0.987074i \(-0.448765\pi\)
0.160266 + 0.987074i \(0.448765\pi\)
\(978\) 0 0
\(979\) −5200.30 −0.169767
\(980\) 0 0
\(981\) 9376.63 0.305171
\(982\) 0 0
\(983\) −4620.40 −0.149917 −0.0749583 0.997187i \(-0.523882\pi\)
−0.0749583 + 0.997187i \(0.523882\pi\)
\(984\) 0 0
\(985\) 3046.29 0.0985410
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6920.73 0.222514
\(990\) 0 0
\(991\) −40365.3 −1.29389 −0.646946 0.762536i \(-0.723954\pi\)
−0.646946 + 0.762536i \(0.723954\pi\)
\(992\) 0 0
\(993\) 19828.5 0.633673
\(994\) 0 0
\(995\) 39167.1 1.24792
\(996\) 0 0
\(997\) 2994.35 0.0951175 0.0475587 0.998868i \(-0.484856\pi\)
0.0475587 + 0.998868i \(0.484856\pi\)
\(998\) 0 0
\(999\) −6641.64 −0.210343
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2352.4.a.cp.1.1 4
4.3 odd 2 1176.4.a.ba.1.1 4
7.3 odd 6 336.4.q.m.289.1 8
7.5 odd 6 336.4.q.m.193.1 8
7.6 odd 2 2352.4.a.cm.1.4 4
28.3 even 6 168.4.q.f.121.1 yes 8
28.19 even 6 168.4.q.f.25.1 8
28.27 even 2 1176.4.a.bd.1.4 4
84.47 odd 6 504.4.s.j.361.4 8
84.59 odd 6 504.4.s.j.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.1 8 28.19 even 6
168.4.q.f.121.1 yes 8 28.3 even 6
336.4.q.m.193.1 8 7.5 odd 6
336.4.q.m.289.1 8 7.3 odd 6
504.4.s.j.289.4 8 84.59 odd 6
504.4.s.j.361.4 8 84.47 odd 6
1176.4.a.ba.1.1 4 4.3 odd 2
1176.4.a.bd.1.4 4 28.27 even 2
2352.4.a.cm.1.4 4 7.6 odd 2
2352.4.a.cp.1.1 4 1.1 even 1 trivial